Large time behavior and pointwise estimates for compressible Euler equations with damping

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• 作者：ZhiGang Wu ; WeiKe Wang
• 关键词：Green’s function ; energy functional ; pointwise estimates ; 35A09 ; 35B40 ; 35J08 ; 35Q35
• 刊名：SCIENCE CHINA Mathematics
• 出版年：2015
• 出版时间：July 2015
• 年：2015
• 卷：58
• 期：7
• 页码：1397-1414
• 全文大小：293 KB
• 参考文献：1.Deng S J, Wang WK. Pointwise decaying rate of large perturbation around viscous shock for scalar viscous conservation law. Sci China Math, 2013, 56: 729-36MATH MathSciNet View Article
  2.Duan R J, Liu H X, Ukai S, et al. Optimal L p-L q convergence rates for the compressible Navier-Stokes equations with potential force. J Differential Equations, 2007, 238: 220-33MATH MathSciNet View Article
  3.Duan R J, Ukai S, Yang T, et al. Optimal convergence rates for the compressible Navier-Stokes equations with potential forces. Math Models Methods Appl Sci, 2007, 17: 737-58MATH MathSciNet View Article
  4.Fang D Y, Xu J. Existence and asymptotic behavior of C1 solutions to the multi-dimensional compressible Euler equations with damping. Nonlinear Anal, 2009, 70: 244-61MATH MathSciNet View Article
  5.Guo Y, Wang Y J. Decay of dissipative equations and negative Sobolev spaces. Comm Partial Differential Equations, 2012, 37: 2165-208MATH MathSciNet View Article
  6.Hsiao L, Liu T P. Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping. Comm Math Phys, 1992, 143: 599-05MATH MathSciNet View Article
  7.Hsiao L, Luo T. Nonlinear diffusive phenomena of solutions for the system of compressible adiabatic flow through porous media. J Differential Equations, 1996, 125: 329-65MATH MathSciNet View Article
  8.Hsiao L, Luo T, Yang T. Global BV solutions of compressible Euler equations with spherical symmetry and damping. J Differential Equations, 1998, 146: 203-25MATH MathSciNet View Article
  9.Hsiao L, Pan R H. Initial boundary value problem for the system of compressible adiabatic flow through porous media. J Differential Equations, 1999, 159: 280-05MATH MathSciNet View Article
  10.Hsiao L, Pan R H. The damped p-system with boundary effects. Contemp Math, 2000, 255: 109-23MathSciNet
  11.Huang F M, Marcati P, Pan R H. Convergence to Barenblatt solution for the compressible Euler equations with damping and vacuum. Arch Ration Mech Anal, 2005, 176: 1-4MATH MathSciNet View Article
  12.Huang F M, Pan R H. Asymptotic behavior of the solutions to the damped compressible Euler equations with vacuum. J Differential Equations, 2006, 220: 207-33MATH MathSciNet View Article
  13.Huang F M, Pan R H. Convergence rate for compressible Euler equations with damping and vacuum. Arch Ration Mech Anal, 2003, 166: 359-76MATH MathSciNet View Article
  14.Huang F M, Pan R H, Wang Z. L 1 convergence to the Barenblatt solution for compressible Euler equations with damping. Arch Ration Mech Anal, 2011, 200: 665-89MATH MathSciNet View Article
  15.Huang F M, Wang Y, Wang Y, et al. Vanishing viscosity of isentropic Navier-Stokes equations for interacting shocks. Sci China Math, 2015, 58: 653-72MathSciNet View Article
  16.Huang M G, Tang M X, Yu J S. Wolbachia infection dynamics by reaction-diffusion equations. Sci China Math, 2015, 58: 77-6MATH MathSciNet View Article
  17.Jiang M N, Zhu C J. Convergence to strong nonlinear diffusion waves for solutions to p-system with damping on quadrant. J Differential Equations, 2009, 246: 50-7MATH MathSciNet View Article
  18.Jiang S, Li F C. Zero dielectric constant limit to the non-isentropic compressible Euler-Maxwell system. Sci China Math, 2014, 57: 2153-162MathSciNet View Article
  19.Li H L, Zhang T. Large time behavior of solutions to 3D compressible Navier-Stokes-Poisson system. Sci China Math, 2012, 55: 159-77MATH MathSciNet View Article
  20.Liao J, Wang W K, Yang T. L p convergence rates of planar waves for multi-dimensional Euler equations with damping. J Differential Equations, 2009, 247: 303-29MATH MathSciNet View Article
  21.Liu T P. Pointwise convergence to shock waves for viscous conservation laws. Comm Pure Appl Math, 1997, 11: 1113-182View Article
  22.Liu T P, Wang W K. The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimension. Comm Math Phys, 1998, 196: 145-73MATH MathSciNet View Article
  23.Liu T P, Zeng Y. Large time behavior of solutions general quasilinear hyperbolic-parabolic systems of conservation laws. Mem Amer Math Soc, 1997
  24.Marcati P, Mei M. Convergence to nonlinear diffusion waves for solutions of the initial boundary value problem to the hyperbolic conservation laws with damping. Quart Appl Math, 2000, 58: 763-83MATH MathSciNet
  25.Marcati P, Pan R H. On the diffusive profiles for the system of compressible adiabatic flow through porous media. SIAM J Math Anal, 2001, 33: 790-26MATH MathSciNet View Article
  26.Nishihara K, Wang W K, Yang T. L p-convergence rate to nonlinear diffusion waves for p-system with damping. J Differential Equations, 2000, 161: 191-18MATH MathSciNet View Article
  27.Nishihara K, Yang T. Boundary effect on asymptotic behavior of solutions to the p-system with damping. J Differential Equations, 1999, 156: 439-58MATH MathSciNet View Article
  28.Pan R H. Darcy’s law as long
• 作者单位：ZhiGang Wu (1)
  WeiKe Wang (2)
  
  1. Department of Applied Mathematics, Donghua University, Shanghai, 201620, China
  2. Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, China
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Chinese Library of Science
  Applications of Mathematics
  
• 出版者：Science China Press, co-published with Springer
• ISSN：1869-1862

文摘

The Cauchy problem of the compressible Euler equations with damping in multi-dimensions is considered when the initial perturbation in H 3-norm is small. First, by using two new energy functionals together with Green’s function and iteration method, we improve the L 2-decay rate in Tan and Wang (2013) and Tan and Wu (2012) when \(\left\| {(\rho _0 - \bar \rho ,m)} \right\|_{\dot B_{1,\infty }^{ - s} \times \dot B_{1,\infty }^{ - s + 1} } \) with s ?[0, 2] is bounded. In particular, it holds that the density converges to its equilibrium state at the rate \((1 + t)^{ - \tfrac{3} {4} - \tfrac{s} {2}} \) in L 2-norm and the momentum decays at the rate \((1 + t)^{ - \tfrac{5} {4} - \tfrac{s} {2}} \) in L 2-norm. Moreover, under a weaker and more general condition on the initial data, we show that the density and the momentum have different pointwise estimates in dimension d with d ?3 on both space variable x and time variable t as \(\left| {D_x^\alpha (\rho - \bar \rho )} \right| \leqslant C(1 + t)^{ - \tfrac{d} {2} - \tfrac{{\left| \alpha \right|}} {2}} (1 + \tfrac{{\left| x \right|^2 }} {{1 + t}})^{ - r} \) with \(r > \tfrac{d} {2} \) and \(\left| {D_x^\alpha m} \right| \leqslant C(1 + t)^{ - \tfrac{d} {2} - \tfrac{{\left| \alpha \right| + 1}} {2}} (1 + \tfrac{{\left| x \right|^2 }} {{1 + t}})^{ - \tfrac{d} {2}} \) by a more elaborate analysis on the Green’s function. These results improve those in Wang and Yang (2001), where the density and the velocity (the momentum) have the same pointwise estimates.